You could try implementing something like this, but I very much suspect it
will be slower this way. For one thing, the space complexity will be
increased by a lot, which is quite significant when you consider how many
rationals might exist at one time. Second, factoring an integer is slow,
even for relatively small integers. On the other hand, the Euclidean
algorithm is very fast, and integer division by the gcd at the end is
extremely fast (one cycle if the integer is small enough). Division would
require a loop in your scheme. The same for multiplication. Addition would
be the worst. You would have do a complete refactorization in general.
Finally, you are mistaken in thinking that 10**20 is sufficient. Internal
calculations of certain algorithms regularly result in very large numbers,
and it's also not uncommon to start with very large numbers in the first
place.
Aaron Meurer
On Feb 28, 2013, at 2:12 PM, Harsh Gupta <gupta.hars...@gmail.com> wrote:
Hi,
I think it would be better if we represent the rational numbers as a
product of primes raised to some integer power.
We can compute and store all the primes we will need to work with a
practically large number, say anyone will not need to work with numbers
larger than 10**20.
Then well will need store only primes less than 10**10, because if some
number is not divisible by any number less than equal to sqrt(n) then it is
prime.
And if we happen to need more primes than we have stored then we can
compute them in the run time.
Representing the rational this way will make multiplication and division of
primes a lot faster.
--
Harsh Gupta
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